Today I proved finiteness for the problem here: Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?
namely: IF we have integers $ \color{red}{x,y,t \geq 1,}$ with $ \color{red}{xy > t}$ and $ \color{red}{q \geq 3}$ such that $$ q = \frac{x^2 + y^2}{xy-t}, $$ THEN $$ q \leq 1 + \frac{324}{25} t^2. $$ That is, for a given value of $t,$ there are finitely many values of $q$ that work.
Given the bounds I found, it is extremely quick to check for solutions by computer. I have done so, and have a simple conjecture, massively komputer konfirmed: $$ \color{red}{ q \leq t^2 + 2 t + 2}. $$
QUESTION: can anyone prove the conjecture? It is true for $t \leq 8192.$
Oh: there is always such a solution for any $t,$ with $x = t+1, y=1.$
The main contribution of the way Hurwitz wrote this sort of thing, pretty much the same idea as Vieta Jumping, is that, if there is any solution with some $t,q,$ then there is a solution with $2x \leq qy$ and $2y \leq qx.$ I have a pdf of his article. The strongest item i got, for such a Hurwitz Grundlösung, was $xy \leq 4t.$ That is mostly from Lagrange multipliers, i think the conjecture needs better, more Diophantine methods.
Here are some samples, because of symmetry I printed only $x \geq y:$
x 2 y 1 t 1 q 5 +++
x 2 y 2 t 2 q 4
x 3 y 1 t 2 q 10 +++
x 2 y 2 t 3 q 8
x 3 y 3 t 3 q 3
x 4 y 1 t 3 q 17 +++
x 4 y 2 t 3 q 4
x 5 y 1 t 3 q 13
x 4 y 2 t 4 q 5
x 5 y 1 t 4 q 26 +++
x 3 y 2 t 5 q 13
x 6 y 1 t 5 q 37 +++
x 7 y 1 t 5 q 25
x 3 y 3 t 6 q 6
x 4 y 2 t 6 q 10
x 7 y 1 t 6 q 50 +++
Why not? here are the 62 fundamental solutions with $x \geq y$ for $t=8192:$
x 96 y 96 t 8192 q 18
x 100 y 84 t 8192 q 82
x 108 y 76 t 8192 q 1090
x 128 y 128 t 8192 q 4
x 152 y 56 t 8192 q 82
x 158 y 54 t 8192 q 82
x 176 y 48 t 8192 q 130
x 188 y 44 t 8192 q 466
x 192 y 64 t 8192 q 10
x 228 y 36 t 8192 q 3330
x 240 y 48 t 8192 q 18
x 241 y 37 t 8192 q 82
x 267 y 31 t 8192 q 850
x 277 y 33 t 8192 q 82
x 288 y 32 t 8192 q 82
x 344 y 24 t 8192 q 1858
x 397 y 21 t 8192 q 1090
x 416 y 32 t 8192 q 34
x 420 y 20 t 8192 q 850
x 440 y 24 t 8192 q 82
x 526 y 22 t 8192 q 82
x 528 y 16 t 8192 q 1090
x 551 y 15 t 8192 q 4162
x 590 y 14 t 8192 q 5122
x 592 y 16 t 8192 q 274
x 633 y 13 t 8192 q 10834
x 661 y 13 t 8192 q 1090
x 684 y 12 t 8192 q 29250
x 716 y 12 t 8192 q 1282
x 796 y 12 t 8192 q 466
x 804 y 20 t 8192 q 82
x 912 y 16 t 8192 q 130
x 913 y 9 t 8192 q 33346
x 1032 y 8 t 8192 q 16642
x 1064 y 8 t 8192 q 3538
x 1171 y 7 t 8192 q 274258
x 1251 y 7 t 8192 q 2770
x 1256 y 8 t 8192 q 850
x 1366 y 6 t 8192 q 466498
x 1374 y 6 t 8192 q 36306
x 1614 y 6 t 8192 q 1746
x 1928 y 8 t 8192 q 514
x 1942 y 6 t 8192 q 1090
x 2052 y 4 t 8192 q 263170
x 2068 y 4 t 8192 q 53458
x 2100 y 4 t 8192 q 21202
x 2196 y 4 t 8192 q 8146
x 2308 y 4 t 8192 q 5122
x 2484 y 4 t 8192 q 3538
x 2759 y 3 t 8192 q 89554
x 2788 y 4 t 8192 q 2626
x 2899 y 3 t 8192 q 16642
x 3303 y 3 t 8192 q 6354
x 3972 y 4 t 8192 q 2050
x 4098 y 2 t 8192 q 4198402
x 4890 y 2 t 8192 q 15058
x 8066 y 2 t 8192 q 8194
x 8193 y 1 t 8192 q 67125250 +++
x 8245 y 1 t 8192 q 1282642
x 8977 y 1 t 8192 q 102658
x 9805 y 1 t 8192 q 59602
x 16257 y 1 t 8192 q 32770